Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Scalar Field in Terms of Two Independent Real Fields

  1. Oct 22, 2005 #1
    I am working with a complex scalar field written in terms of two independent real scalar fields and trying to derive the commutator relations.

    So,

    [tex] \phi = \frac{1}{\sqrt{2}} \left(\phi_1 + i \phi_2) [/tex]

    where [itex]\phi_1[/itex] and [itex]\phi_2[/itex] are real.

    When deriving,

    [tex] [\phi(\vec{x},t),\dot{\phi}(\vec{x}',t)] = 0 [/tex]

    I get terms like the following:

    [tex][\phi_1(\vec{x},t),\dot{\phi}_2(\vec{x}',t)][/tex]

    which I need to vanish. It makes sense to me that they should vanish, but how do I show this?
     
  2. jcsd
  3. Oct 23, 2005 #2
    Hmm...I think that we just take that as the quantization condition. That is,

    [tex]
    [\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t}] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs}
    [/tex]

    Is this correct?
     
  4. Oct 23, 2005 #3

    SpaceTiger

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Since [itex]\phi_1[/itex] and [itex]\phi_2[/itex] are independent, they'll only be canonically conjugate with their own momenta (the [itex]\delta_{rs}[/itex] on the left). Your equation just states that in combination with the usual commutation relation of the real scalar field.
     
  5. Oct 25, 2005 #4
    [tex]\phi_1[/tex] and [tex]\phi_2[/tex]
    are independent fields, so
    [tex][\phi_1, \dot{\phi}_2][/tex]=0
     
  6. Oct 26, 2005 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    What is the Poisson bracket between the classical fields ? If you know that, you can canonically quantize using Dirac's rule.

    Daniel.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Complex Scalar Field in Terms of Two Independent Real Fields
  1. Complex scalar field (Replies: 2)

Loading...